Teaching

• MATH0057 Probability and Statistics - Second term 2020,2021, and 2022

• Aims of course: to introduce students to the theory of probability and some of the statistical methods based upon it. Many real-world processes involve random components which can only be modelled using probabilistic methods. Statistical theory is vital for interpreting information when it is necessary to distinguish genuine patterns from random fluctuations. The course begins with the basic ideas of probability theory: events, probabilities, random variables and the notion of independence. It continues with the two crucial principles: the law of large numbers and the central limit theorem. The course then introduces the fundamental concepts of statistical inference (estimation and hypothesis testing), and illustrates these concepts using the most important statistical models.

• Course contents: Revision of basic ideas in probability (axioms, simple laws, discrete and continuous random variables, expectation). Standard univariate distributions. Joint probability distributions: joint and conditional distributions and moments; serial expectation; multinomial and multivariate normal distributions. Moment and probability generating functions; properties; sums of independent random variables. Statement of weak law of large numbers. Statement of the Central Limit Theorem.

  Introduction to statistics (data, probability models, estimation, hypothesis testing). Normal probability models (1- and 2-sample problems, Chi-squared-distribution, t-distribution, F-distribution). Estimation (point and interval estimation, confidence intervals). Regression and correlation (linear regression, least squares).

• Prerequisites: Good knowledge of calculus and basic combinatorics, as provided by MATH1401 and MATH1201 (for example) and basic probability (the probability content of MATH1301 or equivalent, as taught in post-exam module).

• Texts: comprehensive lecture notes will be made available for this course via Moodle. For students who wish to consult a text in addition to these notes, the recommendation is
  D. Wackerly, W. Mendenhall & R. Scheaffer: Mathematical Statistics with Applications (6th edition), Duxbury Press.
  This text covers all of the material in the course. It contains a large number of worked examples, as well as supplementary exercises for those students who want extra practice (solutions to the supplementary exercises are available in a companion volume).
  The UCL Library stocks a large number of texts on probability and statistics. Two which may be useful for occasional consultation are A First Course in Probability by S.M. Ross, and Introduction to Statistics by R.E. Walpole.

• Assessment: Assessment is an examination in term 3 (90%) (details of which will be decided later), and in-course assessment (10%). The in-course assessment mark will be taken from FOUR of the weekly exercise sheets set during the course: sheets 2, 4, 5, and 8, one of which will be through STACK.

• Other set work: Weekly problem sheets throughout course.

• Timetabled workload: Recorded lectures: About 8 10-to-25-minute videos per week posted on Moodle during term 2 on Monday and Wednesday mornings. One face-to-face teaching per week (Fridays 9-10am or 10-11am) to be used as necessary for feedback on written work and as a problem class as well as for Q&A.

• Moodle: course material will be made available via the course Moodle page. All students taking the course should be enrolled automatically on this Moodle page; if for some reason the course is not showing in your Moodle home page, you should be able to enrol yourself --- just type MATH0057 in the `search courses' box, and use your initiative. Moodle will be the primary method for making course material available: all students are therefore expected to be enrolled on the course Moodle page and to download course materials as appropriate.

• STAT0024 Social Statistics - Second term 2018-2019

• The aim is to provide an introduction to several aspects of sample surveys, namely: survey design and evaluation, statistical sampling theory, measurement in social science and some practical and philosophical aspects.

• On successful completion of the course, a student should have an understanding of the basic principles and methods underlying sample surveys, social measurement and scaling, sampling schemes, and practical survey methods, together with an appreciation of the role of statistics in society and social science.

  Course content: The role of statistics in social science. Planning a survey, questionnaire construction and data collection techniques. Social measurement and scaling. Sources of error, practical survey methods. Basic data presentation. Sampling theory: simple random sampling, stratified sampling, cluster sampling, and systematic sampling. Analysis of social statistics. Methods for handling missing data.

• The course notes will be on the course Moodle page. However, successful participation requires more effort than just 'learning the notes'. Your own thinking and decisions are needed too. Many issues that you will encounter in the real world cannot necessarily be solved by employing a basic toolbox of statistical techniques and learning some facts.

• A 2.5 hour written examination in Term 3, plus a 'take-home' in-course assessment (ICA) given out on 19th February with a deadline of 4pm on 5th March.

• Nine sets of exercises. In each exercise session, we shall discuss the topics and set example questions. You should be prepared to participate in the discussion, either individually or as a group. These will be discussed in problem classes to provide feedback and from time-to-time you may be asked to present your results to the rest of the class, but this will not count towards the examination grading.

• The course notes, examples sheets, ICA information and other material shall be available from the course Moodle page.

• Monte Carlo Inference - Lent 2016

This is the second part of the course "Time Series and Monte Carlo Inference".

• Monte Carlo methods are concerned with the use of stochastic simulation techniques for statistical inference. These have had an enormous impact on statistical practice, especially Bayesian computation, over the last 20 years, due to the advent of modern computing architectures and programming languages. This course covers the theory underlying some of these methods and illustrates how they can be implemented and applied in practice.

• The course will cover the following topics:

  Techniques of random variable generation
  Monte Carlo integration
  Importance Sampling
  Markov chain Monte Carlo (MCMC) methods for Bayesian inference
  Gibbs sampling
  Metropolis-Hastings algorithm
  Applications in network analysis

• Lecture notes will be given on the board. Further notes may be obtained from Alexandra Carpentier's page on this course, written in 2014. However, notice that more material will be covered in our course.

• Two example sheets will be provided and two associated examples classes will be given:

  Example Sheet 1: The associated example class will be held on Thursday 25 February, 9-10am, MR13.

  Example Sheet 2: The associated example class will be held on Friday 22 April, 9-11am, MR13. (Minor edits on the file on 6 March)

• Relationship between distributions (not made by me).

• R codes (Mostly written by Shahin Tavakoli) Data generation , MC integration , Metropolis-Hastings algorithm, Gibbs sampling.

• Further reading:

  P. J. E. Gentle, Random Number Generation and Monte Carlo Methods, (Second Edition). Springer, 2003.

  B. D. Ripley, Stochastic Simulation. Wiley, 1987.

  C.P. Robert and G. Casella, Monte Carlo Statistical Methods, (Second Edition). Springer, 2004.

  C. P. Robert and G. Casella, Introducing Monte Carlo Methods with R. Springer, 2010.

• Graphical Models - Spring 2015

• This course introduces the theory of Graphical Models to graduate students with sufficient mathematical and statistical background. The main goal is to explore the theory behind using the main types of graphical models in the literature, and their relationship to other statistical models and methods used in statistics. An important part of the course will discuss conditional independence in probability distributions and how different types of graphs can capture them. Although, the course, for the most part, will consist of theory, it will also demonstrate through the use of various examples how to fit appropriate models by using known algorithms in the field.

• The course will cover the following topics:

  Graphs and conditional independence
  Markov properties for undirected graphs
  Log-linear models
  Decomposable models
  Markov properties for directed acyclic graphs
  Multivariate regression analysis and graphs
  Gaussian graphical models
  Graphical models for marginal independence
  Markov properties for mixed graphs
  Marginalization and conditioning for DAGs